Functions

The One Thing

A function assigns exactly one output to each allowed input and can be represented by an equation, table, graph, or verbal description. Linear, quadratic, and exponential functions are distinguished by their patterns of change: constant additive change, constant second differences, and constant multiplicative change, respectively.

Definitions and Results

• Function: A relationship in which every input has exactly one output. A relation is a function if no input is paired with more than one output; graphically, this is tested using the vertical line test, in which every vertical line intersects the graph at most once.
• Input and output: The input is the independent quantity, commonly written as $x$. The output depends on the input and is commonly written as $f(x)$ or $y$.
• Domain: The set of possible input values. The domain may be restricted by context; for example, time may be nonnegative and the number of objects may be a whole number.
• Range: The set of output values produced by a function.
• Function notation: Notation such as $f(x)$, read as “$f$ of $x$,” identifying the output for input $x$.
• Linear function: A function with a constant rate of change and a straight-line graph. Its form is

$$f(x)=mx+b,$$ where $m$ is the slope and $b$ is the $y$-intercept. Equal input intervals produce constant first differences.

• Slope: The rate of change of a linear function, calculated as change in output divided by change in input:

$$m=\frac{y_2-y_1}{x_2-x_1},$$ for two points with different $x$-values.

• $y$-intercept: The output when the input is zero. In $y=mx+b$, it is $b$.
• Quadratic function: A function of the form

$$f(x)=a{x}^2+bx+c,$$ where $a\ne 0$. Its graph is a parabola and it has constant second differences for equal input intervals.

• Parabola: The U-shaped or upside-down U-shaped graph of a quadratic function.
• Vertex: The highest or lowest point of a parabola, representing its maximum or minimum. In standard form, its $x$-coordinate is

$$x=-\frac{b}{2a};$$ the corresponding $y$-value is found by substitution.

• Axis of symmetry: The vertical line through the vertex dividing the parabola into two mirror-image halves.
• Quadratic vertex form:

$$f(x)=a(x-h{)}^2+k,$$ with vertex $(h,k)$.

• Quadratic factored form:

$$f(x)=a(x-r_1)(x-r_2),$$ where $r_1$ and $r_2$ are real zeros when they exist.

• Zero or root: An input value making the output zero. On a graph, it is an $x$-intercept. Using the zero product property,

$$(x-r_1)(x-r_2)=0$$ implies $x=r_1$ or $x=r_2$.

• Quadratic formula: The zeros of $a{x}^2+bx+c=0$ are

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

• Discriminant: The quantity $b^2-4ac$. A positive discriminant gives two real zeros, zero gives one repeated zero, and a negative discriminant gives no real zeros.
• Opening of a parabola: $a>0$ produces an upward-opening parabola; $a<0$ produces a downward-opening parabola.
• Exponential function: A function of the form

$$f(x)=a{b}^x,$$ where $a=f(0)$ is the initial value and $b$ is the growth or decay factor.

• Growth factor: The factor by which the output is multiplied for each one-unit increase in input.
• Exponential growth: Occurs when $b>1$.
• Exponential decay: Occurs when $0<b<1$. If a quantity changes by $r$ percent per time period, with $r$ written as a decimal, its factor is $1+r$ for growth and $1-r$ for decay.
• Rate of change: A measure of how much the output changes as the input changes. It may be constant, changing linearly, or changing multiplicatively.
• Average rate of change: Over $[a,b]$,

$$\frac{f(b)-f(a)}{b-a}.$$ For a linear function, this value is the same on every interval.

• Transformation: A graph change caused by shifts, reflections, stretches, or compressions.
• Equivalent forms: Different algebraic expressions representing the same function. For a quadratic, standard form emphasizes the $y$-intercept, factored form emphasizes the zeros, and vertex form emphasizes the vertex and axis of symmetry.
• Piecewise function: A function defined by different rules over different parts of its domain.
• Intersection: A point where two graphs have the same input and output values. It represents a solution to $f(x)=g(x)$.
• Linear versus exponential change: Linear change adds the same amount over equal intervals; exponential change multiplies by the same factor over equal intervals.
• Function comparison: If $f(x)>g(x)$ for a particular input, then the quantity represented by $f$ is greater than that represented by $g$ at that input. Solving $f(x)=g(x)$ identifies equal-output inputs and graphically gives intersection points.

Worked Methods

Identifying whether a relation is a function

• Check whether each input is paired with exactly one output.
• If a relation is represented graphically, apply the vertical line test.
• Conclude that the relation is a function only if no vertical line intersects the graph more than once.
• State the domain and range, taking account of restrictions imposed by the situation.

Building a linear function

• Identify the input and output quantities.
• Determine the initial value, which is the output when the input is zero.
• Determine the constant rate of change.
• Write the function in the form

$$f(x)=mx+b.$$

• If two points are known, calculate the slope using

$$m=\frac{y_2-y_1}{x_2-x_1}.$$

• Check that equal input intervals produce constant first differences.
• Restrict the domain and range if the context requires it.

Recognizing and analyzing a quadratic function

• Identify a quadratic from its standard form

$$f(x)=a{x}^2+bx+c$$ with $a\ne 0$, or from constant second differences in a table.

• Determine the direction of opening from the sign of $a$: $a>0$ opens upward and $a<0$ opens downward.
• Find the vertex $x$-coordinate using

$$x=-\frac{b}{2a}.$$

• Substitute this $x$-value into the function to obtain the vertex $y$-coordinate.
• Identify the axis of symmetry as the vertical line through the vertex.
• Use standard form to identify the $y$-intercept, factored form to identify zeros, or vertex form

$$f(x)=a(x-h{)}^2+k$$ to identify the vertex $(h,k)$.

• Interpret the intercepts, vertex, domain, range, and end behavior using the quantities and units in the context.

Finding quadratic zeros

• If the quadratic is factored, set each factor equal to zero:

$$a(x-r_1)(x-r_2)=0.$$

• Apply the zero product property to obtain

$$x=r_1\text{or}x=r_2.$$

• If factoring is not available, use

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

• Use the discriminant $b^2-4ac$ to determine whether there are two, one repeated, or no real zeros.
• Interpret real zeros as $x$-intercepts or context-specific inputs at which the output is zero.

Building and interpreting an exponential function

• Identify the initial value $a=f(0)$.
• Determine the multiplicative factor $b$.
• Write

$$f(x)=a{b}^x.$$

• For growth, use $b>1$; for decay, use $0<b<1$.
• If the percentage change is $r$, written as a decimal, use $1+r$ for growth or $1-r$ for decay.
• Check that equal input intervals produce the same ratio rather than the same difference.
• Interpret the initial value, factor, domain, range, and long-term behavior in context.

Computing average rate of change

• Select the interval $[a,b]$.
• Evaluate the function at both endpoints.
• Calculate

$$\frac{f(b)-f(a)}{b-a}.$$

• Attach the appropriate units.
• For a linear function, verify that the result is identical on every interval; for nonlinear functions, recognize that it may vary.

Comparing functions

• Express the functions in usable equations, tables, graphs, or verbal descriptions.
• Evaluate both functions at common inputs.
• Compare their values and rates of change.
• Examine graph shape and long-term behavior, not only initial values.
• Solve

$$f(x)=g(x)$$ to find intersection inputs.

• Interpret each intersection as an input where the two functions have equal outputs.
• Use graphing technology or spreadsheets, where appropriate, to display functions, compare values, estimate intersections, and investigate behavior; check all results for reasonable meaning.

Where It Goes Wrong

• Treating a relation as a function without checking that each input has exactly one output; on a graph, the vertical line test must be satisfied.
• Confusing the slope with the $y$-intercept in $f(x)=mx+b$, or calculating slope without ensuring that the two points have different $x$-values.
• Identifying a quadratic from its curved graph alone while forgetting the required form $f(x)=a{x}^2+bx+c$, the condition $a\ne 0$, or the constant-second-difference test.
• Using $-b/(2a)$ as the complete vertex; it gives only the vertex $x$-coordinate, so the corresponding $y$-value must be found by substitution.
• Confusing additive and multiplicative change: linear models add the same amount over equal intervals, whereas exponential models multiply by the same factor; growth requires $b>1$, while decay requires $0<b<1$.
• Reporting graph features without applying contextual restrictions: domain, range, intercepts, turning points, end behavior, and units must make sense for the situation.

What Gets Asked

This material supports questions requiring students to:

• Decide whether a relation or graph represents a function using pairings or the vertical line test.
• Identify inputs, outputs, domain, range, function notation, intercepts, zeros, and units.
• Construct a linear model from a table, graph, two points, or verbal context.
• Calculate slope and average rate of change and interpret them in context.
• Distinguish linear, quadratic, and exponential patterns using first differences, second differences, ratios, equations, tables, and graphs.
• Identify and interpret the standard, factored, and vertex forms of a quadratic.
• Find a quadratic vertex, axis of symmetry, zeros, and opening direction.
• Solve quadratic equations using the zero product property or quadratic formula and interpret the discriminant.
• Construct exponential growth or decay models from an initial value and percentage change.
• Interpret exponential growth factors, decay factors, domain, range, and long-term behavior.
• Apply transformations and recognize equivalent forms.
• Interpret piecewise functions over restricted domains.
• Compare functions by evaluating common inputs, examining tables and graphs, comparing rates of change, and finding intersections through $f(x)=g(x)$.
• Use graphing technology or spreadsheets to estimate intersections and investigate behavior, while checking that the results are mathematically and contextually reasonable.